International Journal of Advances in Science Engineering and Technology, ISSN: 2321-9009
Volume- 2, Issue-4, Oct.-2014
PROFIT FUNCTION OF A TWO-NON IDENTICAL COLD STANDBY
SYSTEM SUBJECT TO FOG WITH SWITCH FAILURE AND REPAIR
FACILITY AS FIRST COME LAST SERVE
ASHOK KUMAR SAINI
Assoc. Prof. (MATHS), Bljs College, Tosham (Bhiwani), India
E-mail: [email protected]
Abstract- To transfer a unit from the standby state to the online state, a device known as ‘switching device’ is required.
Generally, we assume that (1) the switching device is perfect in the sense that it does not fail and (2) the repair facility is first
come first serve basis.. However, there are practical situations where the switching device can also fail and the repair facility
is FCLS i.e First Come Last Serve. We have taken units failure , switch failure distribution as exponential and repair time
distribution as General. We have find out MTSF, Availability analysis, the expected busy period of the server for repair the
failed unit under fog in (0,t], expected busy period of the server for repair in(0,t], the expected busy period of the server for
repair of switch failure in (0,t], the expected number of visits by the repairman for failure of units in (0,t],the expected
number of visits by the repairman for switch failure in (0,t] and Profit benefit analysis using regenerative point technique. A
special case using failure and repair distributions as exponential is derived and graphs have been drawn.
Keyword- Cold Standby, Fog, First Come Last Serve, MTSF, Availability, Busy Period, Profit Function
I.
waits for crossing of other train. When the other train
came it stops and departs first according to FCLS.
INTRODUCTION
To transfer a unit from the standby state to the online
state, a device known as ‘switching device’ is
required. Generally, we assume that the switching
device is perfect in the sense that it does not fail.
However, there are practical situations where the
switching device can also fail.
ASSUMPTIONS
1.The system consists of two identical cold standby
units and the fog and failure time distribution are
exponential with rates λ1, λ2 and λ3 whereas the
repairing rates for repairing the failed system due to
fog and due to switch failure are arbitrary with CDF
G1 (t) & G2 (t) respectively.
2. The operation of units stops automatically when
FOG occurs so that excessive damage of the unit can
be prevented.
3. The Fog actually failed the units. The fog are
non-instantaneous and
it
cannot
occur
simultaneously in both the units.
4. The repair facility works on the come last serve
(FCLS) basis.
5. The switches are imperfect and instantaneous.
6. All random variables are mutually independent.
Symbols for states of the System
Superscripts O, CS, SO, F, SFO
Operative , cold Standby, Stops the operation , Failed,
Switch failed but operable respectively
Subscripts nf, uf,ur, wr, uR
No fog, under fog, under repair, waiting for repair,
under repair continued respectively
This has been pointed out by Gnedenko et
al(1969).Such system in which the switching device
can fail are called systems with imperfect switch. In
the study of redundant systems it is generally
assumed that when the unit operating online fails, the
unit in standby is automatically switched online and
the switchover from standby state to online state is
instantaneous.
In this paper, we have FOG which are noninstantaneous in nature. We assume that the FOG
cannot occur simultaneously in both the units and
when there occurs FOG of the non –instantaneous
nature the operation of the unit stop automatically.
Here, we investigate a two-unit (identical) cold
standby –a system in which offline unit cannot
fail with switch failure under the influence of FOG.
The FOG cannot occur simultaneously in both the
units and when there are less FOG that is within
specifed limit of a unit, it operates as normal as
before but if these are beyond the specified limit the
operation of the unit is avoided and as the FOG goes
on some characteristics of the stopped unit change
which we call failure of the unit. After the FOG are
over the failed unit undergoes repair immediately
according to first come last served discipline. For
example, when a train came on the junction Station it
Up states – 0,1,3
; Down states – 2,4,5,6,7
States of the System
0(Onf , CSnf)
One unit is operative and the other unit is cold
standby and there are no fog in both the units.
1(SOnf , Onf)
Profit Function of a Two-Non Identical Cold Standby System Subject to Fog with Switch Failure and Repair Facility as First Come Last Serve
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International Journal of Advances in Science Engineering and Technology, ISSN: 2321-9009
The operation of the first unit stops automatically due
to fog and cold standby units starts operating.
2(SOnf , SFOnf,ur)
The operation of the first unit stops automatically due
to fog and during switchover to the second unit
switch fails and undergoes repair..
3(Fur , Ouf)
p43+ P43
(5)
Volume- 2, Issue-4, Oct.-2014
=1
(2)
And mean sojourn time are
µ0 = E(T) =
= -1/ λ1
Similarly
The first one unit fails and undergoes repair after the
fog are over and the other unit continues to be
operative with no fog.
4(Fur , SOuf)
The one unit fails and undergoes repair after the fog
are over and the other unit also stops automatically
due to fog.
µ1
=
1/ λ2 ,
µ2 =
,
µ4 =
(3)
Mean Time To System Failure
We can regard the failed state as absorbing
5(FuR , Fwr)
,
The repair of the first unit is continued from state 4
and the other unit is failed due to fog in it & is
waiting for repair.
(4-6)
Taking Laplace-Stieltjes transforms of eq. (4-6) and
solving for
6(FuR , SOuf)
=
The repair of the first unit is continued from state 3
and in the other unit fog occur and stops
automatically due to fog.
7(Fwr , SFOuR)
Where
N1(s) =
yield
(7)
{
+
D1(s) = 1 Making use of relations (1) & (2) it can be shown that
=1, which implies
The repair of failed switch is continued from state 2
and the first unit is failed after fog and waiting for
repair.
Transition Probabilities
Simple probabilistic considerations
following expressions:
N1(s) / D1(s)
that
the
is a proper distribution.
MTSF = E[T] =
=
p01 =
, p02 =
p13 =
, p14 =
= (
*
p23= λ1G2 ( λ2) ,
p24=
p23
+ p01 p13
)
(8)
*
= λ2G2 ( λ2) ,
( λ2) ,
p30= G1*( λ1) , p33(6)=
+p01
/(1 - p01 p13 p30 )
(7)
*
2
s=0
(D1’(0) - N1’(0)) / D1 (0)
where
+
+
+
+
*
1
( λ1)
p43 = G1*( λ2) , p43(5) = G1*( λ2)
(1)
,
(1)
+
+
,
,
(6)
(5)
Availability analysis
Let Mi(t) be the probability of the system having
started from state i is up at time t without making any
other regenerative state belonging to E. By
probabilistic arguments, we have tThe value of
We can easily verify that
p01 + p02 = 1, p13 + p14 = 1, p23 + p23(1) + p24= 1 ,
p30 + p33(6) = 1,
M0(t)=
λ1
λ3 t , M1(t)=
λ1
λ2 t
Profit Function of a Two-Non Identical Cold Standby System Subject to Fog with Switch Failure and Repair Facility as First Come Last Serve
64
International Journal of Advances in Science Engineering and Technology, ISSN: 2321-9009
λ1
M3(t)
).
1
Volume- 2, Issue-4, Oct.-2014
(9)
(t) =
The steady state availability
A0 =
So that
(16)
The expected Busy period of the server for repair of
switch in (o,t]
=
=
Using L’ Hospitals rule, we get
A0 =
In the long run, P0 =
where N5(0)= p30 p02
defined.
=
(10)
0(0)
+ p01
1(0)
3(0)
2(0)
and D2 (0) is already
The expected busy period of the server for repair of
the switch in (0,t] is
Where
N2(0)= p30
(17)
’
)
(t) =
’
D2 (0) =
+ p01
+ p14
+
p02(
+ p24
)] p30
The expected up time of the system in (0,t] is
So
that
io\(18)
The expected number of visits by the repairman for
repairing the different units in (0,t]
(t) =
So that
(11)
The expected down time of the system in (0,t] is
(t) = t(t)
So that
In the long run , H0 =
(19)
where N6(0)= p30 and D’3(0) is already defined.
The expected number of visits by the repairman for
repairing the switch in (0,t]
(12)
In the long run, V0 =
(20)
where N7(0)= p30 [p01 p14 p43 + p02 ] and D’3(0) is
already defined.
Cost Benefit Analysis
The cost-benefit function of the system considering
mean up-time, expected busy period of the system
under fog when the units stops automatically,
expected busy period of the server for repair of unit
& switch, expected number of visits by the repairman
for unit failure, expected number of visits by the
repairman for switch failure.
The expected busy period of the server for repairing
the failed unit under fog in (0,t]
In the long run,
R0 =
(13)
where N3(0)= p30 [ p01 (
1(0)
+ p14
4(0)
) + p02
( 2(0) + p24 4(0))
and D2’(0) is already defined.
The expected period of the system under fog in (0, t]
is
(t)
=
So
The expected total cost per unit time in steady state is
C=
=
= K1A0 - K2P0 - K3B0 - K4R0 - K5V0 - K6H0
that
(14)
The expected Busy period of the server for repair of
dissimilar units by the repairman in (0,t]
In steady state, B0 =
Where
K1 - revenue per unit up-time,
K2 - cost per unit time for which the system is under
switch repair
K3 cost per unit time for which the system is
under unit repair
K4 cost per unit time for which the system is
under fog when units automatically stop.
K5 cost per visit by the repairman for which
switch repair, K6 - cost per visit by the repairman
for units repair.
(15)
where N4(0)= 3(0)+ 4(0) { p30 (p01p14 + p02 p24) }
and D2’(0) is already defined.
The expected busy period of the server for repair in
(0,t] is
Profit Function of a Two-Non Identical Cold Standby System Subject to Fog with Switch Failure and Repair Facility as First Come Last Serve
65
International Journal of Advances in Science Engineering and Technology, ISSN: 2321-9009
CONCLUSION
After studying the system, we have analysed
graphically that when the failure rate, fog rate
increases, the MTSF and steady state availability
decreases and the cost function decreased as the
failure increases.
REFERENCES
[1]
Volume- 2, Issue-4, Oct.-2014
[2]
Dhillon, B.S. and Natesen, J, Stochastic Anaysis of outdoor
Power Systems in fluctuating environment, Microelectron.
Reliab. .1983;23, 867-881.
[3]
Gnedanke, B.V., Belyayar, Yu.K. and Soloyer , A.D. ,
Mathematical Methods of Relability Theory, 1969 ;
Academic Press, New York.
[4]
Goel, L.R., Sharma,G.C. and Gupta, Rakesh Cost Analysis
of a Two-Unit standby system with different weather
conditions, Microelectron. Reliab. ,1985; 25, 665-659.
[5]
Goel,L.R. ,Sharma G.C. and Gupta Parveen , Stochastic
Behaviour and Profit Anaysis of a redundant system with
slow switching device, Microelectron Reliab., 1986; 26,
215-219.
Barlow, R.E. and Proschan, F., Mathematical theory of
Reliability, 1965; John Wiley, New York.
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Profit Function of a Two-Non Identical Cold Standby System Subject to Fog with Switch Failure and Repair Facility as First Come Last Serve
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